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equation of a circle
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Definition/Summary
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| A circle has many definitions, a classical one being "the locus of all points on a plane that are equidistant from a given point, which is referred to as the 'center' of the circle". |
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Equations
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Equation for a circle with it's center as the origin and radius 'r':
[tex]
x^2 + y^2 = r^2
[/tex]
Equation for a circle with center at (a, b) and radius 'r':
[tex]
(x - a)^2 + (y - b)^2 = r^2
[/tex]
General equation (expanded) for a circle:
[tex]
x^2 + y^2 + 2fx + 2gy + c = 0
[/tex]
The center of such a circle is given as (-f, -g) and the radius 'r' is given by:
[tex]
r = \sqrt{g^2 + f^2 - c}
[/tex]
The slope of the tangent at a point 'x' for a circle centered at (a, b) is given as:
[tex]
\tan \theta = -\frac{x - a}{y - b}
[/tex]
Area of a circle is given as:
[tex]
A = \pi r^2
[/tex]
Length of an arc subtending an angle [itex]\theta[/itex] is given as:
[tex]
L = r\theta
[/tex]
For the length of the complete circle, [itex]\theta = 2\pi[/itex]
For a circle, centered at origin, and radius 'r' a point on the circle, the radius to which makes an angle [itex]\theta[/itex] with the positive x-axis is given as:
[tex]
x = r\sin \theta
[/tex]
[tex]
y = r\cos \theta
[/tex] |
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Recent forum threads on equation of a circle
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Breakdown
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Mathematics
> Geometry
>> Coordinate Geometry
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Extended explanation
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Other definitions from more general concepts:
1. a special instance of an ellipse, having an eccentricity, [itex]e = 0[/itex] (i.e., equal lengths of major and minor axes),
2. a conic section formed by the intersection of a cone with a plane normal to its axis of symmetry,
3. the two-dimensional instance of an n-dimensional hypersphere. |
Commentary
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